TEST IIMath 232March 18, 2004 Name:By writing my name I swear by the honor code.Read all of the following information before starting the exam:• Show all work, clearly and in order. You will not get full credit if I cannot see how you arrivedat your answer (even if your final answer is correct).• Make sure that you follow the directions in each problem and that your answer matches whatis asked for.• Justify your answers algebraically whenever possible. For most problems, work done bycalculator will not receive any points (although you may use your calculator to check youranswers).• Please keep your written answers brief; be clear and to the point. I will take points off forrambling and for incorrect or irrelevant statements.• By writing your name above, you agree to the JMU honor code. In particular, this meansthat you may not use any notes or crib sheets during this exam, that all work must be yourown, and that you may not obtain advance information revealing the problems on this exam.• This test has 8 problems and is worth 100 points, plus some extra credit at the end. Makesure that you have all of the pages!• Good luck!1. (12 pts) Determine whether each of the following statements is true (T) or false (F).(a) T F For all real numbers x we have 1 + cot2x = csc2x.(b) T F If θ is an integer multiple of π, then cos θ = −1.(c) T F The period of f (x) = tan x is π.(d) T F If sec x = 2 then cos−1x =12.(e) T F A limit of the form ∞ − ∞ is in an indeterminate form.(f) T F A limit of the form 0∞is in an indeterminate form.2. (10 pts) Give short answers.(a) Give a formal definition of sin θ, for any angle θ. Your definition should includethe words “unit circle,” “standard position,” “terminal edge,” and “coordinate.”You can illustrate your definition with a picture if it makes it clearer.(b) Sketch a clear, labeled graph of f(x) = sec−1x. (It may help to begin by sketchingthe graph of the restricted secant function; if you do this, make sure it is clearwhich graph is which.)3. (10 pts) Prove thatddx(ln x) =1x. (Hint: Start with the fact that eln x= x for all x > 0.)4. (10 pts) Prove thatddx(cot x) = − csc2x. (Hint: Use the derivatives of sin x and cos x.)5. (24 pts) Fill in the blanks.(a) limx→0+log3x =(b)ddx(ln |x|) =(c) If limx→0ln(f(x)) = −∞, then limx→0f(x) =(d) If θ is an angle in standard position whose terminal edge intersects the unit circleat the point (−14,√154), then tan θ =(e) limx→∞sin xex=(f) limx→∞exsin x =(g) limx→0+x1 − cos x=(h) limx→2sin(x2− 4)x2− 4=(i) limx→∞arctan x =(j)ddx(sec−1x) =(k) Suppose f(x) is a general sine function with period 4π and a minimum at x =π2.The first maximum to the right of this minimum occurs at x =(l) Two points for writing anything you want, plus two extra points for writing the“secret word” from the policy sheet:6. (10 pts) Use logarithmic differentiation to find the derivative of f (x) = xsin x. Show your workvery carefully, and circle your final answer.7. (24 pts) Fill in the blanks.(a) The domain of f(x) = csc x is(b) The range of f(x) = csc x is(c) The domain of f(x) = cos−1x is(d) The range of f(x) = cos−1x is(e) sin(sin−1x) = x for all x in the interval(f) The graph of the function f(x) = −4 sin(3(x − π)) + 2 has:amplitude: period: ce nter point:(g) If sin−1x = θ and s in θ is negative, then θ is in the quadrant.(h) List all the x-values of the inflection points of f(x) = cos x:Survey Questions: (2 extra credit points)Name a question or topic that could have been on this test, but wasn’t.How do you think you did?SPACE FOR SCRAP